Group Theory of the Periodic Table - Welcome to SCIPPscipp.ucsc.edu/~haber/ph251/Group Theory of the...
Transcript of Group Theory of the Periodic Table - Welcome to SCIPPscipp.ucsc.edu/~haber/ph251/Group Theory of the...
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Group Theory of the Periodic Table
John Tamanas
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Periodic TableA periodic system that classifies the chemical elements
Proposed by Dmitrii Mendeleev on Feb 17, 1869
Bill Gates’s Periodic Table
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QM of Periodic System1. Principal and orbital quantum numbers (n,l) characterize orbitals2. Pauli Exclusion Principle3. Aufbau Principle - Electrons fill up energy levels in order of increasing energy.
These are the properties we’ll want to include in our description of the periodic table.
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Derivation of Group StructureWe’ll want to be able to exhibit/explain
1. Period doubling2. Madelung rule
We’ll do this by using the non-invariance group SO(4,2) of the hydrogenic orbitals
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Period DoublingThe periods of the periodic table are of length 2 - 8 - 8 - 18 - 18 - 32 - 32 …
This is not trivial. For example, if electrons filled up atoms in terms of increasing n, the periods would be 2 - 8 - 18 - 32...
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Madelung Rule“With increasing nuclear charge Z, one-electron orbitals are filled according to increasing N=n+l. For fixed N, the orbitals are filled in order of increasing n.”
This is why Potassium’s electron configuration, for example, is [Ne]3s23p64s1 rather than 3s23p63d1 that one would guess if orbitals were filled in terms of increasing n.
This naturally gives rise to the 2-8-8-18-18-... periodicity doubling.
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Correlation diagram for different orbital filling rules
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SO(4,2)⊗SU(2)Chemistry is determined by electromagnetic interactions.
SO(4,2) is isomorphic to the conformal group in Minkowski space.
SO(4,2) is the maximal dynamical non-invariance group for a hydrogen-like atom.
SU(2) only handles spins, so we’ll mainly focus on SO(4,2).
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Non-invariance groupSome of the generators of the Lie algebra will no longer commute with the Hamiltonian.
We will be generating spectra, which are infinite. Therefore, we will be considering a non-compact non-invariance group
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Atomic ChessKing
Queen
Rook
Bishop
Knight
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Atomic ChessboardAll elements belong to an (n,l) configuration on the chessboard.
Each square represents 2(2l+1) substates
n = 1,..., ∞
l = 0,...,n-1
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King ♚ and Queen ♛These can travel in all directions along the chessboard
Their moves are therefore allowed by SO(4,2)
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The Rook ♜ (Horizontal moves)Moving horizontally on the chessboard means moving over states of different l, but same n.
Thinking of the correlation diagram, this is like the n-rule for orbital filling.
Explains “accidental symmetry” of n=2 Hydrogen.
Corresponds to dynamical SO(4).
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The Rook ♜ (Horizontal moves)Moving horizontally on the chessboard means moving over states of different l, but same n.
Thinking of the correlation diagram, this is like the n-rule for orbital filling.
Explains “accidental symmetry” of n=2 Hydrogen.
Corresponds to SO(4) dynamical symmetry.
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The Rook ♜ (Vertical moves)Moving vertically with fixed l is the same as moving radially
Therefore, vertical moves correspond to raising and lower operators of the radial group, SO(2,1).
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The Knights ♞(n-l/2) (main knight)
● Connects (n,l) with (n+1,l+2) and (n-1,l-2).
● Identified with SU(3).
(n+l/2) (second knight)
● Connects (n,l) with (n+1,l-2) and (n-1,l+2).
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The Knights ♞Main knight gives Cartan-Weyl diagrams
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The Bishops ♝Madelung (n+l)
● Associated with ╲ diagonals
Regge (n-l)
● Associated with ╱ diagonals
Can be identified with SO(3,2), the anti-De Sitter group.
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Left Step Periodic Table
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So what about period doubling?The bishops, which represent SO(3,2) symmetries, thus divide SO(4,2) into two sets: n+l even and n+l odd.
And so the period doubling we see (2-18-18-32-32-...) is due to SO(4,2) breaking into SO(3,2).
Under SO(3,2) the manifold splits into two subsets.
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Raising Bishops to OperatorsWe introduce from O(4,2).
This operator does not commute with all SO(4,2) operators. Not a bad thing!
With and B, one can define
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Regge Operators
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Madelung Operators
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RecapWe took SO(4,2) then broke it into SO(3,2).
We then realized we could break this further into two subsets, one for each Bishop.
So we introduced a new operator to create SO’(3,1) (tilted SO(3,1)).
Now have an algebra so(4,2)-like algebra with Madelung and Regge operators.
Let’s look at algebraic structure.
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Algebraic StructureNot very different from standard SO(4), SO(3,1) algebras.
Because the structure constants are operators (and not constants), this is a nonlinear algebra.
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Casimir Operatorsso(3,1) is an algebra of order r = 6 and rank l = 2. By Racah’s theorem, there are two Casimir operators in each group.
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Regge Algebra
Casimir OperatorsMadelung Algebra
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ConclusionWe played some atomic chess and realized some hidden group theoretic structure in the periodic table.
The group that best describes the elements is SO(4,2)⊗SU(2).
“Atoms are Christian because they listen to bishops.”
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References1. A. Ceulemans and P. Thyssen. “Shattered Symmetry: Group Theory from the
Eightfold way to the Periodic Table”.
2. M. Kibler. “A Group-Theoretical Approach to the Periodic Table of Chemical Elements: Old and New Developments”. arxiv:quant-ph/0503039
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Thank you